I want to achieve an effect simmmilar to one in THIS video. Basically I want to animate the scale of the specific (e.g. top ones) parts of objects' "layers" created via Array modifier as well as the z value of them, so they appear as they are not all the same distance one from another...
Using an empty as Object offset to control the array modifier will give you a lot of options. If you play with the reltive offset you can get things like this:
Changing the distance and size of the empty you'll control the relative distance of the cloned items in the array. You can then add keyframes:
By adding a lattice on the scene and a lattice modifier to the plane to the stack you can further control the deformation (as in the opening of the video used as reference on the question: